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Edited By Team Careers360 | Updated on Jan 29, 2024 10:45 AM IST

Addition and subtraction are the basic arithmetic operations in mathematics. Fractions are the values that denote the parts of a whole thing. Generally, a fraction is formatted so that it has a numerator and a denominator. Here, the denominator represents the total number of parts of a whole thing, and the numerator denotes the required part or parts from the total number of parts. Based on this, the fractions are categorised into three types. They are proper, improper, and mixed fractions. So, applying basic operations like addition and subtraction to fractions is a little different from the general numerals. The addition and subtraction operations on fractions are explained by using models in pre-primary and primary class students. Here the definition of these operations is the same as the basic but the only difference is the process.

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Definition: Fractions are a numerical representation of parts of a whole thing or a number of equal parts. A fraction is itself a division. The numerator of the fraction acts as the dividend and the denominator acts as the divisor.

Fractions are of three types or three kinds. They are proper fractions, improper fractions, and mixed fractions.

Proper fractions: These are the fractions whose numerator value is less than the denominator value.

Examples: \frac{1}{2} or \frac{5}{6}; here the denominator is greater than the numerator.

Improper fractions: These are the fractions whose numerator value is greater than the denominator value.

Examples: \frac{3}{2} or \frac{11}{9}; here the denominator is less than the numerator.

Mixed fractions: These are the fractions that have a quotient and remainder. In these, a quotient is a whole number and the remainder is the numerator of the fraction. So, the mixed fraction is the combination of a whole number and a proper fraction.

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Download E-bookExample: \[5\frac{1}{2}\];

The addition of fractions is as simple as the addition of general numbers. But the only thing is here we have numerators and denominators. If the denominators are the same for the fractions which are to be added, then only the numerators are added and the denominator remains the same.

For example:

Consider two fractions with the same denominator. Such as and

For adding these two fractions, first, we check the denominators. If they are the same then, the process of addition becomes too simple. That is, we need to add only the numerators in the result and the denominator of the result is the same as those fractions.

\frac{4}{5}+\frac{3}{5}=\frac{4+3}{5}=\frac{7}{5}

Subtraction of fractions is similar to the process of the addition of fractions with the same denominator.

For example:

Consider two fractions with the same denominator. Such as and

Then, the subtraction is as follows:

\frac{4}{5}-\frac{3}{5}=\frac{4-3}{5}=\frac{1}{5}

While adding or subtracting fractions, we first need to check the denominators. If they are not the same, then we need to reframe or rewrite those fractions into the same denominators. For this, we need to multiply a number by the denominator as well as by the numerator of one of the fractions or both fractions to make the same denominators.

For example:

Consider two fractions. Such as ; \frac{3}{5} and ; \frac{1}{4}. Here, the denominators are different. So, we need to frame them into the same value. So, by multiplying the numerator and the denominator of both the fractions with the denominator of the other fraction. This is also said to be the process of finding the least common multiple (LCM) of the denominators. I.e.,

The fraction ; \frac{3}{5} is multiplied by the denominator of the other fraction ; \frac{1}{4}.

So, we can write ; \frac{3\times 4}{5\times 4}=\frac{12}{20}

The fraction ; \frac{1}{4} is multiplied by the denominator of the other fraction ; \frac{3}{5}.

So, we can write ; \frac{1\times 5}{4\times 5}=\frac{5}{20}

Then, the required subtraction of the fractions becomes

; \frac{3}{5}+\frac{1}{4}=\frac{12}{20}+\frac{5}{20}

Since the denominators of both fractions are the same, the numerators are to be added. I.e.,

\[\begin{align}

& \frac{12}{20}+\frac{5}{20}=\frac{12+5}{20}=\frac{17}{20} \\

& \therefore \frac{3}{5}+\frac{1}{4}=\frac{17}{20} \\

\end{align}\]

The subtraction of fractions with different denominators is also as similar to the addition of fractions with different denominators.

For example:

Consider two fractions. Such as ; \frac{2}{3} and ; \frac{1}{5}.

The fraction ; \frac{2}{3} is multiplied by the denominator of the other fraction ; \frac{1}{5}.

So, we can write ; \frac{2\times 5}{3\times 5}=\frac{10}{15}

The fraction ; \frac{1}{5} is multiplied by the denominator of the other fraction ; \frac{2}{3}.

So, we can write \frac{1\times 3}{5\times 3}=\frac{3}{15}

Then, the required subtraction of the fractions becomes

; \frac{2}{3}-\frac{1}{5}=\frac{10}{15}-\frac{3}{15}

Since the denominators of both fractions are the same, the numerators are to be added. I.e.,

\[\begin{align}

& \frac{10}{15}-\frac{3}{15}=\frac{10-3}{15}=\frac{7}{15} \\

& \therefore \frac{2}{3}-\frac{1}{5}=\frac{7}{15} \\

\end{align}\]

Mixed fractions are the combination of proper fractions and whole numbers. To add this type of mixed fraction, first, we need to convert them into an improper fraction i.e., we need to combine the whole number and proper fraction into a fraction. Then, we need to check the denominators. If the denominators are the same then we need to add the numerators. Otherwise use the above process of adding fractions with different denominators. For converting the mixed fractions, we need to multiply the denominator with the whole number and then add the numerator to that value and place the whole result in the numerator and the denominator will be the same as the mixed fraction.

For example: Consider two mixed fractions as 2\frac{1}{4} and 3\frac{1}{4}.

On simplifying the mixed fractions into improper fractions, we get

; 2\frac{1}{4}=\frac{2\times 4+1}{4}=\frac{9}{4}

; 3\frac{1}{4}=\frac{3\times 4+1}{4}=\frac{13}{4}

Then, the addition of these fractions is

; 2\frac{1}{4}+3\frac{1}{4}=\frac{9}{4}+\frac{13}{4}

Since the denominators are the same, we need to add the numerators.

\begin{align}

& \frac{9}{4}+\frac{13}{4}=\frac{22}{4} \\

& \therefore 2\frac{1}{4}+3\frac{1}{4}=\frac{22}{4} \\

\end{align}

Since the obtained fraction is an improper fraction, we need to convert it into a mixed fraction as

\frac{22}{4}=\frac{11}{2}=5\frac{1}{2}

Where 5 is the quotient and 1 is the remainder.

Thus, the result of the addition is ; 2\frac{1}{4}+3\frac{1}{4}=5\frac{1}{2}.

Subtraction operation on mixed fractions is also similar to the addition operation on mixed fractions. The steps to be followed are:

First, we need to convert the mixed fraction into an improper fraction.

Then, subtract the obtained fractions by a process according to the denominator.

If the resultant fraction is an improper fraction, we can convert it into the mixed form of the fraction by simple division method. Otherwise, leave it as it is.

For example:

Consider two mixed fractions as 2\frac{1}{7} and 1\frac{1}{2}.

On simplifying the mixed fractions into improper fractions, we get

2\frac{1}{7}=\frac{2\times 7+1}{7}=\frac{15}{7}

1\frac{1}{2}=\frac{1\times 2+1}{2}=\frac{3}{2}

Then, the subtraction of these fractions is

2\frac{1}{7}-1\frac{1}{2}=\frac{15}{7}-\frac{3}{2}

Since the denominators are different, we need to frame them into the same value by using multiplication as mentioned above.

\frac{15}{7}-\frac{3}{2}=\frac{15\times 2}{7\times 2}-\frac{3\times 7}{2\times 7}=\frac{30}{14}-\frac{21}{14}

Now, the denominators of the fractions are the same, so, we can calculate their subtractions as

\begin{align}

& \frac{30}{14}-\frac{21}{14}=\frac{30-21}{14}=\frac{9}{14} \\

& \therefore 2\frac{1}{7}-1\frac{1}{2}=\frac{9}{14} \\

\end{align}

Since the obtained fraction is a proper fraction, we can not convert it into a mixed fraction.

1. What is a fraction?

A fraction denotes a part of a whole or a number of equal parts. It has a numerator and denominator.

2. Write about mixed fractions.

Mixed fractions consist of a quotient and a remainder. The quotient is represented by a whole number and the remainder is given in the numerator of the proper fraction. So, by this, we can say that a mixed fraction is the combination of a whole number and a proper fraction.

3. If the denominator is greater than the numerator, then is it a valid fraction?

Yes, it is a valid fraction and it is said to be a proper fraction. Example: \frac{1}{2}.

4. How to add fractions with the same denominators?

Consider two fractions with the same denominators as \frac{1}{2} and \frac{3}{2}.

Then their addition is ; \frac{1}{2}+\frac{3}{2}=\frac{1+3}{2}=\frac{4}{2}.

5. Do we use LCM in the addition and subtraction of fractions?

Yes, we use LCM in the addition and subtraction of fractions with different denominators. Since the denominators are different we need to multiply the least common multiple by both fractions.

Addition and Subtraction of Fractions

Apr 27, 2022 - 12:42 p.m. IST ---STATIC

Addition And Subtraction

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